The equation of motion for a spring-mass system excited by a harmonic force is

                                       Mẍ + Kx = F cos(ωt),

where M is the mass, K is the spring stiffness, F is the force amplitude and ω is the angular frequency of excitation. Resonance occurs when ω is equal to

The correct answer is:

$$\sqrt{\frac{K}{M}}$$

Step-by-Step Explanation:

1. Understanding the Equation of Motion:
The equation of motion for a single-degree-of-freedom, undamped spring-mass system subjected to a harmonic excitation force is given by:

$$M\stackrel{··}{x}+Kx=F\cos \left(\omega t\right)$$
where:
- M represents the mass of the system,
- K represents the spring stiffness,
- F is the amplitude of the exciting force,
- ω is the angular frequency of the exciting force, and
- $\stackrel{··}{x}$ represents the acceleration of the mass (the second time-derivative of displacement x).

2. Finding the Natural Frequency:
The natural angular frequency of the system, denoted as ${\omega }_n$, is the frequency at which the system naturally vibrates when free from any external forces. We can find it by considering the homogeneous (free vibration) equation:

$$M\stackrel{··}{x}+Kx=0$$
Dividing the entire equation by mass M gives:

$$\stackrel{··}{x}+\frac{K}{M}x=0$$
Comparing this with the standard equation of simple harmonic motion, $\stackrel{··}{x}+{\omega }_n^{2}x=0$, we identify:

$${\omega }_n^2=\frac{K}{M}$$
Taking the square root of both sides, the natural angular frequency is:

$${\omega }_n=\sqrt{\frac{K}{M}}$$

3. Understanding Resonance:
Resonance is a physical phenomenon that occurs when the frequency of the externally applied harmonic force (ω) matches the natural frequency of vibration of the system (${\omega }_n$). Under this condition, the energy transferred to the system is maximized, and for an undamped system, the amplitude of vibration theoretically increases to infinity.
Thus, resonance occurs when:

$$\omega ={\omega }_n=\sqrt{\frac{K}{M}}$$
This corresponds to the expression shown in the correct option.